Developable Sculptural Forms of Ilhan Koman Tevfik Akgun¨ Ahmet Koman Ergun Akleman Faculty of Art and Design Molecular Biology and Visualization Sciences Program, Design Communication Genetics Department Department of Architecture Department Bogazic¸i˜ University Texas A&M University Yildiz Technical University Istanbul, Turkey College Station, Texas, USA Istanbul, Turkey [email protected] [email protected] [email protected] Abstract Ilhan Koman was one of the innovative sculptors of the 20th century [9, 10]. He frequently used mathematical concepts in creating his sculptures and discovered a wide variety of sculptural forms that can be of interest for the art+math community. In this paper, we focus on developable sculptural forms he invented approximately 25 years ago, during a period that covers the late 1970’s and early 1980’s. 1 Introduction π + π + π + π + π+ Hyperform Figure 1: The photograph π +π +π +π +π+ series is from a Koman exhibition at Gronningen, Copenhagen (Denmark) in 1986. Hyperform is realized in stainless steel for the steel company Atlas Copco, (Entrance hall, Atlas Copco, Sweden, 1973) (Photos by Koman family) In this paper, we want to present developable forms invented by sculptor Ilhan Koman during the 1970’s. The first is the π + π + π + π + π+ a series of forms derived by the increase of the surface of a circle without changing its radius. A second one is called rolling lady and created using four identical cones. The third is the ”Hyperform” that he produced by self-joining a single sheet of metal. Extension of the length of the rectangular sheet in a defined manner led to multiple derivatives. Figure 1 shows photographs of π + π + π + π + π+ and Hyperform developable sculptures. 2 Developable Surfaces Developable surfaces are defined as the surfaces on which the Gaussian curvature is 0 everywhere [20]. The developable surfaces are useful since they can be made out of sheet metal or paper by rolling a flat sheet of material without stretching it [14]. Most large-scale objects such as airplanes or ships are constructed using un-stretched sheet metals, since sheet metals are easy to model and they have good stability and vibration properties. Moreover, sheet metals provide good fluid dynamic properties. In ship or airplane design, the problems usually stem from engineering concerns and in engineering design there has been a strong interest in developable surfaces. For instance, modeling packages such as Rhino provides developable surface analysis [14, 15]. Although, once invented, it is easy to physically construct developable surfaces using sheet metal or paper, it is not that easy to provide computational models to represent developable surfaces. Sun and Fiume developed a technique for constructing developable surfaces [19], but, their method is useful only to represent ribbons and it is hard to use to represent general developable surfaces. Haeberli has recently introduced a method to represent a shape with piecewise developable surfaces and developed a Lamina Design Software [7]. The current results seem to be limited but the Haeberli’s approach Lamina has a great potential for developable surface design. Mitani and Suzuki introduced a method to approximate any given shape using developable surfaces to create paper models [11]. Because of the approximate nature of their models, there exists gaps between individual pieces and therefore, their method is not suitable for engineering application. Sheet metal is not only excellent for stability, fluid dynamics and vibration, but also one can construct aesthetic buildings and sculptures using sheet metal or paper. Developable surfaces are frequently used by contemporary architects, allowing them to design new forms. However, the design and construction of large- scale shapes with developable surfaces requires extensive architectural and civil engineering expertise. Only a few architectural firms such as Gehry Associates can take advantage of the current graphics and modeling technology to construct such revolutionary new forms [12]. Some architectural structures can be easier to construct with developable surfaces. For instance, Fishback and Tuazon introduced Randome, a dome like structure that is constructed from developable surfaces [6]. Developable surfaces are particularly interesting for sculptural design. It is possible to find new forms by physically constructing developable surfaces. Recently, very interesting developable sculptures, called D-forms, were invented by the London designer Tony Wills and introduced by Sharp, Pottman and Wallner [16, 13]. D-forms are created by joining the edges of a pair of sheet metal or paper with the same perimeter [16, 13]. Pottman and Wallner introduced two open questions involving D-forms [13, 5]. Sharp introduced anti-D-forms that are created by joining the holes [17]. Ron Evans invented another related developable form called Plexagons [4]. Paul Bourke has recently constructed computer generated both D-forms and plexons [2, 4] using Evolver developed by Ken Brakke [1]. 3 Ilhan Koman Biography Ilhan Koman was born in 1921, Edirne, Turkey, studied at the Art Academy in Istanbul, opened his first work- shop and exhibition in Paris, 1948, moved to Sweden in 1958, where he taught at the Konstfack School of Applied Art in Stockholm until his death in 1986. Figure 2 shows two photographs of Ilhan Koman. At right Koman is photographed in front of his and architect Chet Kanra’s sculpture, called “From Leonardo To...” in Stockholm. His works cover a wide spectrum of styles and materials, including 12 public monuments in Sweden and 4 in Turkey. He is represented in several museums including Moderna Museet, Stockholm, Museum of Modern Art, New York, Musee´ d’Art Moderne de la Ville de Paris. During the last twenty years of his life, he worked on inventing diverging geometrical forms which he developed as prototypes to be real- ized in large-scale projects. Among the geometrical shapes he developed, tetraflex was a flexible polyhedron which he registered at the Swedish Patent Office in 1971. At the time he had created and proposed these Figure 2: Two photographs of Ilhan Koman (left photo by Christer Strmholm, 1970’s; right photo by Tayfun Tunc¸elli, 1980’s). structures as modules for architecture, constructions in space and aviation fuel tanks [9, 10] 4 Vertex Angle Deflections and π + π + π + π + π+ Form Ilhan Koman’s developable sculptures in π + π + π + π + π+ form [8, 10] vividly illustrate vertex angle de- flections which are less known but very useful mathematical devices to understand local behavior of surfaces [3]. Vertex deflection is a measure to show how much a given region is deflected from plane. In practice, local behavior can be changed by vertex deflections with nip and tuck. For instance, By simply take out some angle (nipping or pinching) we can create a convex or concave region with paper (See Figure 3). To create saddle regions we can add angles to a planar paper surface by tucking as shown in Figure 3. Initial Paper Maxima Saddle Figure 3: Creation of maxima and saddle with subtracting and adding angles to a flat surface. To construct π + π + π + π + π+ sculptures, Koman started from a circular sheet of plastic or metal and added angles (tucking) by constructing a series of wild saddle structures by forcing the angle deflection in a given vertex much smaller than −2π [8, 10]. As shown in Figure 4, when the angle decreases lower than −2π, it is possible to create a wide variety of saddle shapes. We observe that Koman’s developable structures can give very useful visual intuition about the local behavior at a vertex for piecewise linear meshes. Vertex angle deflections can formally defined around vertices of surfaces that are piecewise linear or piecewise developable. A mesh M is called piecewise linear if every edge of M is a straight line segment, and every face of M is planar. Piecewise linear meshes are useful since the surface of each face is well-defined [?]. Most common piecewise linear meshes are triangular meshes. However, piecewise linear meshes are a significant generalization of triangular meshes, which not only allow non-triangular faces, but also allow faces to have different face valence. Figure 4: Ilhan Koman’s Developable Sculptures. These were constructed with sheet metal in the 1980’s. In these sculptures, the connections are almost invisible (photos by Tayfun Tunc¸elli). Vertex angle deflections are defined based on the corner angles around a vertex (µi) as mi θ¯(µi) = 2π − ∑ θ j j=1 where θ j is the internal angle of the j-th corner of vertex µi and mi is the valence of the vertex µi. Our recent result show that sum of vertex angle deflections v ∑ θ¯(µi) = 2π(2 − 2g) i=1 where g is the genus (number of holes and handles) and 2−2g is the Euler Characteristic of the surface. This result is in sync with Gauss-Bonnet theorem [21] which states that the integral of the Gaussian curvature over a closed smooth surface is equal to 2π times the Euler characteristic the surface. Vertex angle deflection can provide a good intuition about surface curvature. It is interesting to point out that the value of θ¯(µi) is somewhat related to curvature and, in fact, Calladine used vertex angle deflections to estimate curvature in triangular meshes [3]. • θ¯(µi) > 0: then vertex µi is either convex or concave. • θ¯(µi) = 0: then vertex µi is planar. • θ¯(µi) < 0: then vertex µi is a saddle point. Ilhan Koman’s sculptures are easy to make and they can be great tool to teach the effect of vertex angle deflections. Koman was planning to create an extremely large version of one of his wild π +π +π +π +π+ with a platform in the center such that people will go the center of the resulting spherical shape will only see space curve.